A uniform circular track is the area bounded by two concentric circles. If the ares of the track is 1144 `m^(2)` and its width is 14 m, find the diameters of the two circles.

Answer

To solve the problem of finding the diameters of two concentric circles that form a uniform circular track, we will follow these steps:

### Step 1: Understand the Problem
We have a circular track formed by two concentric circles. The area of the track is given as 1144 m², and the width of the track is 14 m. We need to find the diameters of both circles.

### Step 2: Define Variables
Let:
- \( r \) = radius of the inner circle
- \( R \) = radius of the outer circle

Since the width of the track is 14 m, we can express the radius of the outer circle in terms of the inner circle's radius:
\[ R = r + 14 \]

### Step 3: Area of the Track
The area of the track is the difference between the area of the outer circle and the area of the inner circle:
\[ \text{Area of the track} = \pi R^2 - \pi r^2 = 1144 \]
Factoring out \(\pi\), we have:
\[ \pi (R^2 - r^2) = 1144 \]

### Step 4: Substitute for R
Substituting \( R = r + 14 \) into the area equation:
\[ \pi ((r + 14)^2 - r^2) = 1144 \]

### Step 5: Expand the Equation
Expanding \((r + 14)^2\):
\[ (r + 14)^2 = r^2 + 28r + 196 \]
Thus, we have:
\[ \pi (r^2 + 28r + 196 - r^2) = 1144 \]
This simplifies to:
\[ \pi (28r + 196) = 1144 \]

### Step 6: Solve for r
Dividing both sides by \(\pi\):
\[ 28r + 196 = \frac{1144}{\pi} \]
Using \(\pi \approx 3.14\):
\[ 28r + 196 \approx \frac{1144}{3.14} \approx 364.24 \]
Now, subtract 196 from both sides:
\[ 28r \approx 364.24 - 196 \approx 168.24 \]
Dividing by 28:
\[ r \approx \frac{168.24}{28} \approx 6.01 \text{ m} \]
For practical purposes, we can round \( r \) to 6 m.

### Step 7: Find R
Now, we can find \( R \):
\[ R = r + 14 = 6 + 14 = 20 \text{ m} \]

### Step 8: Calculate Diameters
Finally, we calculate the diameters of both circles:
- Diameter of the inner circle:
\[ \text{Diameter}_{\text{inner}} = 2r = 2 \times 6 = 12 \text{ m} \]
- Diameter of the outer circle:
\[ \text{Diameter}_{\text{outer}} = 2R = 2 \times 20 = 40 \text{ m} \]

### Final Answer
- Diameter of the inner circle: **12 m**
- Diameter of the outer circle: **40 m**

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